I'm using a binomial logistic regression to identify if exposure to has_x
or has_y
impacts the likelihood that a user will click on something. My model is the following:
fit = glm(formula = has_clicked ~ has_x + has_y,
data=df,
family = binomial())
This the output from my model:
Call:
glm(formula = has_clicked ~ has_x + has_y,
family = binomial(), data = active_domains)
Deviance Residuals:
Min 1Q Median 3Q Max
0.9869 0.9719 0.9500 1.3979 1.4233
Coefficients:
Estimate Std. Error z value Pr(>z)
(Intercept) 0.504737 0.008847 57.050 < 2e16 ***
has_xTRUE 0.056986 0.010201 5.586 2.32e08 ***
has_yTRUE 0.038579 0.010202 3.781 0.000156 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 217119 on 164182 degrees of freedom
Residual deviance: 217074 on 164180 degrees of freedom
AIC: 217080
Number of Fisher Scoring iterations: 4
As each coefficient is significant, using this model I'm able to tell what the value of any of these combinations is using the following approach:
predict(fit, data.frame(has_x = T, has_y=T), type = "response")
I don't understand how I can report on the Std. Error of the prediction.

Do I just need to use $1.96*SE$? Or do I need to convert the
$SE$ using an approach described here? 
If I want to understand the standarderror for both variables
how would I consider that?
Unlike this question, I am interested in understanding what the upper and lower bounds of the error are in a percentage. For example, of my prediction shows a value of 37% for True,True
can I calculate that this is $+/ 0.3%$ for a $95\% CI$? (0.3% chosen to illustrate my point)
Best Answer
Your question may come from the fact that you are dealing with Odds Ratios and Probabilities which is confusing at first. Since the logistic model is a non linear transformation of $\beta^Tx$ computing the confidence intervals is not as straightforward.
Background
Recall that for the Logistic regression model
Probability of $(Y = 1)$: $p = \frac{e^{\alpha + \beta_1x_1 + \beta_2 x_2}}{1 + e^{ \alpha + \beta_1x_1 + \beta_2 x_2}}$
Odds of $(Y = 1)$: $ \left( \frac{p}{1p}\right) = e^{\alpha + \beta_1x_1 + \beta_2 x_2}$
Log Odds of $(Y = 1)$: $ \log \left( \frac{p}{1p}\right) = \alpha + \beta_1x_1 + \beta_2 x_2$
Consider the case where you have a one unit increase in variable $x_1$, i.e. $x_1 + 1$, then the new odds are
$$ \text{Odds}(Y = 1) = e^{\alpha + \beta_1(x_1 + 1) + \beta_2x_2} = e^{\alpha + \beta_1 x_1 + \beta_1 + \beta_2x_2 } $$
$$ \frac{\text{Odds}(x_1 + 1)}{\text{Odds}(x_1)} = \frac{e^{\alpha + \beta_1(x_1 + 1) + \beta_2x_2} }{e^{\alpha + \beta_1 x_1 + \beta_2x_2}} = e^{\beta_1} $$
Log Odds Ratio = $\beta_1$
Relative risk or (probability ratio) = $\frac{ \frac{e^{\alpha + \beta_1x_1 + \beta_1 + \beta_2 x_2}}{1 + e^{ \alpha + \beta_1x_1 + \beta_1 + \beta_2 x_2}}}{ \frac{e^{\alpha + \beta_1x_1 + \beta_2 x_2}}{1 + e^{ \alpha + \beta_1x_1 + \beta_2 x_2}}}$
Interpreting coefficients
How would you interpret the coefficient value $\beta_j$ ? Assuming that everything else remains fixed:
Confidence intervals for a single parameter $\beta_j$
Since the parameter $\beta_j$ is estimated using Maxiumum Likelihood Estimation, MLE theory tells us that it is asymptotically normal and hence we can use the large sample Wald confidence interval to get the usual
$$ \beta_j \pm z^* SE(\beta_j)$$
Which gives a confidence interval on the logodds ratio. Using the invariance property of the MLE allows us to exponentiate to get $$ e^{\beta_j \pm z^* SE(\beta_j)}$$
which is a confidence interval on the odds ratio. Note that these intervals are for a single parameter only.
If you include several parameters you can use the Bonferroni procedure, otherwise for all parameters you can use the confidence interval for probability estimates
Bonferroni procedure for several parameters
If $g$ parameters are to be estimated with family confidence coefficient of approximately $1  \alpha$, the joint Bonferroni confidence limits are
$$ \beta_g \pm z_{(1  \frac{\alpha}{2g})}SE(\beta_g)$$
Confidence intervals for probability estimates
The logistic model outputs an estimation of the probability of observing a one and we aim to construct a frequentist interval around the true probability $p$ such that $Pr(p_{L} \leq p \leq p_{U}) = .95$
One approach called endpoint transformation does the following:
Since $Pr(x^T\beta) = F(x^T\beta)$ is a monotonic transformation of $x^T\beta$
$$ [Pr(x^T\beta)_L \leq Pr(x^T\beta) \leq Pr(x^T\beta)_U] = [F(x^T\beta)_L \leq F(x^T\beta) \leq F(x^T\beta)_U] $$
Concretely this means computing $\beta^Tx \pm z^* SE(\beta^Tx)$ and then applying the logit transform to the result to get the lower and upper bounds:
$$[\frac{e^{x^T\beta  z^* SE(x^T\beta)}}{1 + e^{x^T\beta  z^* SE(x^T\beta)}}, \frac{e^{x^T\beta + z^* SE(x^T\beta)}}{1 + e^{x^T\beta + z^* SE(x^T\beta)}},] $$
The estimated approximate variance of $x^T\beta$ can be calculated using the covariance matrix of the regression coefficients using
$$ Var(x^T\beta) = x^T \Sigma x$$
The advantage of this method is that the bounds cannot be outside the range $(0,1)$
There are several other approaches as well, using the delta method, bootstrapping etc.. which each have their own assumptions, advantages and limits.
Sources and info
My favorite book on this topic is "Applied Linear Statistical Models" by Kutner, Neter, Li, Chapter 14
Otherwise here are a few online sources:
Edit October 2021  New links